We study the problem of extracting randomness from somewhere random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections.

A somewhere-random source is a tuple (X_1, \ldots, X_t) of (possibly correlated) \{0,1\}^n-valued random variables X_i where for some unknown i \in [t], X_i is guaranteed to be ... more >>>